One-sided convergence in noncommutative individual ergodic theorems

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, or, more generally, in a noncommutative Orlicz space, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $\mu_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $\mu_t(x)$ is a non-increasing rearrangement of $x$.